Evolutionary dynamics of the continuous iterated prisoner's dilemma

The iterated prisoner's dilemma (IPD) has been widely used in the biological and social sciences to model dyadic cooperation. While most of this work has focused on the discrete prisoner's dilemma, in which actors choose between cooperation and defection, there has been some analysis of the continuous IPD, in which actors can choose any level of cooperation from zero to one. Here, we analyse a model of the continuous IPD with a limited strategy set, and show that a generous strategy achieves the maximum possible payoff against its own type. While this strategy is stable in a neighborhood of the equilibrium point, the equilibrium point itself is always vulnerable to invasion by uncooperative strategies, and hence subject to eventual destabilization. The presence of noise or errors has no effect on this result. Instead, generosity is favored because of its role in increasing contributions to the most efficient level, rather than in counteracting the corrosiveness of noise. Computer simulation using a single-locus infinite alleles Gaussian mutation model suggest that outcomes ranging from a stable cooperative polymorphism to complete collapse of cooperation are possible depending on the magnitude of the mutational variance. Also, making the cost of helping a convex function of the amount of help provided makes it more difficult for cooperative strategies to invade a non-cooperative equilibrium, and for the cooperative equilibrium to resist destabilization by non-cooperative strategies. Finally, we demonstrate that a much greater degree of assortment is required to destabilize a non-cooperative equilibrium in the continuous IPD than in the discrete IPD. The continuous model outlined here suggests that incremental amounts of cooperation lead to rapid decay of cooperation and thus even a large degree of assortment will not be sufficient to allow cooperation to increase when cooperators are rare. The extreme degree of assortment required to destabilize the non-cooperative equilibrium, as well as the instability of the cooperative equilibrium, may help explain why cooperation in Prisoner's Dilemmas is so rare in nature.     

Evolutionary escape from the prisoner's dilemma

The classic prisoner's dilemma model of game theory is modified by introducing occasional variations on the options available to players. Mutation and selection of game options reliably change the game matrix, gradually, from a prisoner's dilemma game into a byproduct mutualism one, in which cooperation is stable, and "temptation to defect" is replaced by temptation to cooperate. This result suggests that when there are many different potential ways of interacting, exploring those possibilities may make escape from prisoner's dilemmas a common outcome in the world. A consequence is that persistent prisoner's dilemma structures may be less common than one might otherwise expect.     

Evolutionary game dynamics in a Wright-Fisher process

Evolutionary game dynamics in finite populations can be described by a frequency dependent, stochastic Wright-Fisher process. We consider a symmetric game between two strategies, A and B. There are discrete generations. In each generation, individuals produce offspring proportional to their payoff. The next generation is sampled randomly from this pool of offspring. The total population size is constant. The resulting Markov process has two absorbing states corresponding to homogeneous populations of all A or all B. We quantify frequency dependent selection by comparing the absorption probabilities to the corresponding probabilities under random drift. We derive conditions for selection to favor one strategy or the other by using the concept of total positivity. In the limit of weak selection, we obtain the 1/3 law: if A and B are strict Nash equilibria then selection favors replacement of B by A, if the unstable equilibrium occurs at a frequency of A which is less than 1/3.     

Evolutionary dynamics in structured populations

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called ‘spatial selection’: cooperators prevail against defectors by clustering in physical or other spaces.     

Evolutionary dynamics of prey exploitation in a metapopulation of predators

In well-mixed populations of predators and prey, natural selection favors predators with high rates of prey consumption and population growth. When spatial structure prevents the populations from being well mixed, such predators may have a selective disadvantage because they do not make full use of the prey's growth capacity and hence produce fewer propagules. The best strategy then depends on the degree to which predators can monopolize the exploitation of local prey populations, which in turn depends on the spatial structure, the number of migrants, and, in particular, the stochastic nature of the colonization process. To analyze the evolutionary dynamics of predators in a spatially structured predator-prey system, we performed simulations with a metapopulation model that has explicit local dynamics of nonpersistent populations, keeps track of the number of emigrants entering the migration pool, assumes individuals within local populations as well as within the migration pool to be well mixed, and takes stochastic colonization into account. We investigated which of the predator's exploitation strategies are evolutionarily stable and whether these strategies minimize the overall density of prey, as is the case in Lotka-Volterra-type models of competitive exclusion. This was analyzed by pairwise invasibility plots based on short-term simulations and tested by long-term simulation experiments of competition between resident and mutant predator-types that differed in one of the following parameters: the prey-to-predator conversion efficiency, the per capita prey consumption rate, or the per capita emigration rate from local populations. In addition, we asked which of these three strategies are most likely to evolve. Our simulations showed that under selection for conversion efficiency the predator-prey system always goes globally extinct yet persists under selection for consumption or emigration rates and that the evolutionarily stable (ES) exploitation strategies do not maximize local population growth rates. The most successful exploitation strategy minimizes the overall density of prey but does not make it settle exactly at the minimum. The system did not settle at the point where the mean time to co-invasion (i.e., immigration of a second predator in a local prey population) equals the mean local interaction time (an idea borne out from studies on host exploitation strategies in host-pathogen systems) but rather where the mean time to co-invasion was larger. The ES exploitation strategies represent more prudent strategies than the ones that minimize prey density. Finally, we show that-compared to consumption-emigration is a more likely target for selection to achieve prudent exploitation and that prudent exploitation strategies can evolve only provided the prey-to-predator conversion efficiency is subject to constraints.     

Evolutionary game theory,interpersonal comparisons and natural selection: a dilemma

When social scientists began employing evolutionary game theory (EGT) in their disciplines, the question arose what the appropriate interpretation of the formal EGT framework would be. Social scientists have given different answer, of which I distinguish three basic kinds. I then proceed to uncover the conceptual tension between the formal framework of EGT, its application in the social sciences, and these three interpretations. First, I argue that EGT under the biological interpretation has a limited application in the social sciences, chiefly because strategy replication often cannot be sensibly interpreted as strategy bearer reproduction in this domain. Second, I show that alternative replication mechanisms imply interpersonal comparability of strategy payoffs. Giving a meaningful interpretation to such comparisons is not an easy task for many social situations, and thus limits the applicability of EGT in this domain. Third, I argue that giving a new interpretation both to strategy replication and selection solves the issue of interpersonal comparability, but at the costs of making the new interpretation incompatible with natural selection interpretations of EGT. To the extent that social scientists seek such a natural selection interpretation, they face a dilemma: either face the challenge that interpersonal comparisons pose, or give up on the natural selection interpretation. By identifying these tensions, my analysis pleas for greater awareness of the specific purposes of EGT modelling in the social sciences, and for greater sensitivity to the underlying microstructure on which the evolutionary dynamics and other EGT solution concepts supervene.     

Evolutionary dynamics of transposable elements in a small RNA world

Transposable elements (TEs) are selfish elements that cause harmful mutations, contribute to the structure of regulatory networks and shape the architecture of genomes. Natural selection against their harmful effects has long been considered the dominant force limiting their spread. It is now clear that a genome defense system of RNA-mediated silencing also plays a crucial role in limiting TE proliferation. A full understanding of TE evolutionary dynamics must consider how these forces jointly determine their proliferation within genomes. Here I consider these forces from two perspectives - dynamics within populations and evolutionary games within the germline. The analysis of TE dynamics from these two perspectives promises to provide new insight into their role in evolution.     

Evolutionary stability in strategic models of single-locus frequency-dependent viability selection

The dynamic stability of an evolutionarily stable strategy (ESS) is analyzed for a diploid species under individual viability selection. An individual's viability depends on the genotypic frequencies at a single autosomal locus through a payoff matrix determined by phenotypic behaviours (i.e. strategies). It is shown that an ESS of this payoff matrix is dynamically stable if there are at most three alleles — an intuitive result that strengthens the importance of static game-theoretic methods in genetic models.Author for correspondence     

Evolutionary dynamics of taxonomic structure

The distribution of species among genera and higher taxa has largely untapped potential to reveal among-clade variation in rates of origination and extinction. The probability distribution of the number of species within a genus is modelled with a stochastic, time-homogeneous birth-death model having two parameters: the rate of species extinction, μ, and the rate of genus origination, γ, each scaled as a multiple of the rate of within-genus speciation, λ. The distribution is more sensitive to γ than to μ, although μ affects the size of the largest genera. The species : genus ratio depends strongly on both γ and μ, and so is not a good diagnostic of evolutionary dynamics. The proportion of monotypic genera, however, depends mainly on γ, and so may provide an index of the genus origination rate. Application to living marine molluscs of New Zealand shows that bivalves have a higher relative rate of genus origination than gastropods. This is supported by the analysis of palaeontological data. This concordance suggests that analysis of living taxonomic distributions may allow inference of macroevolutionary dynamics even without a fossil record.     

Evolutionary dynamics of microsatellite DNA

Within the past decade microsatellites have developed into one of the most popular genetic markers. Despite the widespread use of microsatellite analysis, an integral picture of the mutational dynamics of microsatellite DNA is just beginning to emerge. Here, I review both generally agreed and controversial results about the mutational dynamics of microsatellite DNA. Microsatellites are short DNA sequence stretches in which a motif of one to six bases is tandemly repeated. It has been known for some time that these sequences can differ in repeat number among individuals. With the advent of polymerase chain reaction (PCR) technology this property of microsatellite DNA was converted into a highly versatile genetic marker (Litt and Luty 1989; Tautz 1989; Weber and May 1989). Polymerase chain reaction products of different length can be amplified with primers flanking the variable microsatellite region. Due to the availability of high-throughput capillary sequencers or mass spectrography the sizing of alleles is no longer a bottleneck in microsatellite analysis. The almost random distribution of microsatellites and their high level of polymorphism greatly facilitated the construction of genetic maps (Dietrich et al. 1994; Dib et al. 1996) and enabled subsequent positional cloning of several genes. Almost at the same time, microsatellites were established as the marker of choice for the identification of individuals and paternity testing. The high sensitivity of PCR-based microsatellite analysis was not only of great benefit for forensics, but opened completely new research areas, such as the analysis of samples with limited DNA amounts (e.g., many social insects) or degraded DNA (e.g., feces, museum material) (Schl?tterer and Pemberton 1998). More recently, microsatellite analysis has also been employed in population genetics (Goldstein and Schl?tterer 1999). Compared with allozymes, microsatellites offer the advantage that, in principle, several thousand potentially polymorphic markers are available. Nevertheless, the application of microsatellites to population genetic questions requires a more detailed understanding of the mutation processes of microsatellite DNA as the evolutionary time frames covered in population genetics are often too long to allow novel microsatellite mutations to be ignored. Additional interest in the evolution of microsatellite DNA comes from the discovery that trinucleotide repeats, a special class of microsatellites, are involved in human neurodegenerative diseases (e.g., fragile X and Huntington's disease). A detailed understanding of the processes underlying microsatellite instability is therefore an important contribution toward a better understanding of these human neurodegenerative diseases.     

Evolutionary dynamics of Ralstonia solanacearum

We investigated the genetic diversity, extent of recombination, natural selection, and population divergence of Ralstonia solanacearum samples obtained from sources worldwide. This plant pathogen causes bacterial wilt in many crops and constitutes a serious threat to agricultural production due to its very wide host range and aggressiveness. Five housekeeping genes, dispersed around the chromosome, and three virulence-related genes, located on the megaplasmid, were sequenced from 58 strains belonging to the four major phylogenetic clusters (phylotypes). Whereas genetic variation is high and consistent for all housekeeping loci studied, virulence-related gene sequences are more diverse. Phylogenetic and statistical analyses suggest that this organism is a highly diverse bacterial species containing four major, deeply separated evolutionary lineages (phylotypes I to IV) and a weaker subdivision of phylotype II into two subgroups. Analysis of molecular variations showed that the geographic isolation and spatial distance have been the significant determinants of genetic variation between phylotypes. R. solanacearum displays high clonality for housekeeping genes in all phylotypes (except phylotype III) and significant levels of recombination for the virulence-related egl and hrpB genes, which are limited mainly to phylotype strains III and IV. Finally, genes essential for species survival are under purifying selection, and those directly involved in pathogenesis might be under diversifying selection.     

Evolutionary game dynamics in finite populations

We introduce a model of stochastic evolutionary game dynamics in finite populations which is similar to the familiar replicator dynamics for infinite populations. Our focus is on the conditions for selection favoring the invasion and/or fixation of new phenotypes. For infinite populations, there are three generic selection scenarios describing evolutionary game dynamics among two strategies. For finite populations, there are eight selection scenarios. For a fixed payoff matrix a number of these scenarios can occur for different population sizes. We discuss several examples with unexpected behavior.     

Evolutionary dynamics of intratumor heterogeneity

Intraneoplastic diversity in human tumors is a widespread phenomenon of critical importance for tumor progression and the response to therapeutic intervention. Insights into the evolutionary events that control tumor heterogeneity would be a major breakthrough in our comprehension of cancer development and could lead to more effective prevention methods and therapies. In this paper, we design an evolutionary mathematical framework to study the dynamics of heterogeneity over time. We consider specific situations arising during tumorigenesis, such as the emergence of positively selected mutations ("drivers") and the accumulation of neutral variation ("passengers"). We perform exact computer simulations of the emergence of diverse tumor cell clones over time, and derive analytical estimates for the extent of heterogeneity within a population of cancer cells. Our methods contribute to a quantitative understanding of tumor heterogeneity and the impact of heritable alterations on this tumor trait.     

Evolutionary dynamics of biological auctions

Many scenarios in the living world, where individual organisms compete for winning positions (or resources), have properties of auctions. Here we study the evolution of bids in biological auctions. For each auction, n individuals are drawn at random from a population of size N. Each individual makes a bid which entails a cost. The winner obtains a benefit of a certain value. Costs and benefits are translated into reproductive success (fitness). Therefore, successful bidding strategies spread in the population. We compare two types of auctions. In “biological all-pay auctions”, the costs are the bid for every participating individual. In “biological second price all-pay auctions”, the cost for everyone other than the winner is the bid, but the cost for the winner is the second highest bid. Second price all-pay auctions are generalizations of the “war of attrition” introduced by Maynard Smith. We study evolutionary dynamics in both types of auctions. We calculate pairwise invasion plots and evolutionarily stable distributions over the continuous strategy space. We find that the average bid in second price all-pay auctions is higher than in all-pay auctions, but the average cost for the winner is similar in both auctions. In both cases, the average bid is a declining function of the number of participants, n. The more individuals participate in an auction the smaller is the chance of winning, and thus expensive bids must be avoided.     

Evolutionary dynamics of habitat use

I examine the evolution of alternate genotypes that use two habitats that differ in vegetative cover, focusing on the interplay between ecological dynamics of the community and changes in selective advantage. Facultative habitat choice can stabilize a predator population that would cycle if isolated in the more open habitat. This has important implications for the evolution of habitat use strategies. Local stability arising from facultative habitat use allows any number of behavioural genotypes to co-exist: selective use of the open habitat, selective use of the dense habitat, opportunistic use of both habitats in proportion to availability, and facultative switching between habitats to maximize energy gain. Co-existence occurs because the fitness landscape is flat at the ecological equilibrium imposed by the facultative genotype. In contrast, ecological instability favours the evolution of genotypes with behavioural flexibility to avoid being in the wrong place at the wrong time or selective exploitation of one of the habitats. Uncertain information about habitat quality erodes the adaptive advantage of otherwise optimal behaviours, favouring a bet-hedging behavioural strategy synonymous with partial habitat preferences. These results suggest that ecological dynamics could have a strong influence on behavioural heterogeneity within forager populations and that a mixed ESS for habitat use should predominate.     

Evolutionary dynamics of viral attenuation

The genetic trajectory leading to viral attenuation was studied in a canine parvovirus (CPV) strain grown on dog kidney cells for 115 transfers. Consensus sequences of viral populations at passages 0, 3, 30, 50, 80, and 115 were obtained from PCR products covering 86% of the genome; clones from each of the 80th and 115th passages were also sequenced, covering 69% of the genome. Sixteen changes were fixed in the 115th-passage virus sample. Levels of polymorphism were strikingly different over time, in part because of a plaque-cloning step at passage 112 that reduced variation: passage 80 had 19 variants common among the clones, but passage 115 had only a single common variant. Several mutations increased in the culture at the same time, with most reaching fixation only after the 80th passage. The pattern of evolution was consistent with recombination and not with separate selective sweeps of individual mutations. Thirteen of the changes observed were identical to or at the same positions as changes observed in other isolates of CPV or feline panleukopenia virus.     

Evolutionary dynamics of zero-sum games

Aim model in terms of differential equations is used to explain mammalian ovulation control, in particular regulation for a prescribed number of mature eggs. NIH Grant RO1 GM 32153-01GE